Algebra Examples

$x^{5} - 3 x^{4} - 15 x^{3} + 45 x^{2} - 16 x + 48 = 0$

Step 1

Factor the left side of the equation.

Tap for more steps...

Step 1.1

Regroup terms.

$x^{5} - 15 x^{3} - 16 x - 3 x^{4} + 45 x^{2} + 48 = 0$

Step 1.2

Factor $x$ out of $x^{5} - 15 x^{3} - 16 x$ .

Tap for more steps...

Step 1.2.1

Factor $x$ out of $x^{5}$ .

$x \cdot x^{4} - 15 x^{3} - 16 x - 3 x^{4} + 45 x^{2} + 48 = 0$

Step 1.2.2

Factor $x$ out of $- 15 x^{3}$ .

$x \cdot x^{4} + x (- 15 x^{2}) - 16 x - 3 x^{4} + 45 x^{2} + 48 = 0$

Step 1.2.3

Factor $x$ out of $- 16 x$ .

$x \cdot x^{4} + x (- 15 x^{2}) + x \cdot - 16 - 3 x^{4} + 45 x^{2} + 48 = 0$

Step 1.2.4

Factor $x$ out of $x \cdot x^{4} + x (- 15 x^{2})$ .

$x (x^{4} - 15 x^{2}) + x \cdot - 16 - 3 x^{4} + 45 x^{2} + 48 = 0$

Step 1.2.5

Factor $x$ out of $x (x^{4} - 15 x^{2}) + x \cdot - 16$ .

$x (x^{4} - 15 x^{2} - 16) - 3 x^{4} + 45 x^{2} + 48 = 0$

Step 1.3

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x ({(x^{2})}^{2} - 15 x^{2} - 16) - 3 x^{4} + 45 x^{2} + 48 = 0$

Step 1.4

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x (u^{2} - 15 u - 16) - 3 x^{4} + 45 x^{2} + 48 = 0$

Step 1.5

Factor $u^{2} - 15 u - 16$ using the AC method.

Tap for more steps...

Step 1.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 16$ and whose sum is $- 15$ .

$- 16, 1$

Step 1.5.2

Write the factored form using these integers.

$x ((u - 16) (u + 1)) - 3 x^{4} + 45 x^{2} + 48 = 0$

Step 1.6

Replace all occurrences of $u$ with $x^{2}$ .

$x ((x^{2} - 16) (x^{2} + 1)) - 3 x^{4} + 45 x^{2} + 48 = 0$

Step 1.7

Rewrite $16$ as $4^{2}$ .

$x ((x^{2} - 4^{2}) (x^{2} + 1)) - 3 x^{4} + 45 x^{2} + 48 = 0$

Step 1.8

Factor.

Tap for more steps...

Step 1.8.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$x ((x + 4) (x - 4) (x^{2} + 1)) - 3 x^{4} + 45 x^{2} + 48 = 0$

Step 1.8.2

Remove unnecessary parentheses.

$x (x + 4) (x - 4) (x^{2} + 1) - 3 x^{4} + 45 x^{2} + 48 = 0$

Step 1.9

Factor $3$ out of $- 3 x^{4} + 45 x^{2} + 48$ .

Tap for more steps...

Step 1.9.1

Factor $3$ out of $- 3 x^{4}$ .

$x (x + 4) (x - 4) (x^{2} + 1) + 3 (- x^{4}) + 45 x^{2} + 48 = 0$

Step 1.9.2

Factor $3$ out of $45 x^{2}$ .

$x (x + 4) (x - 4) (x^{2} + 1) + 3 (- x^{4}) + 3 (15 x^{2}) + 48 = 0$

Step 1.9.3

Factor $3$ out of $48$ .

$x (x + 4) (x - 4) (x^{2} + 1) + 3 (- x^{4}) + 3 (15 x^{2}) + 3 (16) = 0$

Step 1.9.4

Factor $3$ out of $3 (- x^{4}) + 3 (15 x^{2})$ .

$x (x + 4) (x - 4) (x^{2} + 1) + 3 (- x^{4} + 15 x^{2}) + 3 (16) = 0$

Step 1.9.5

Factor $3$ out of $3 (- x^{4} + 15 x^{2}) + 3 (16)$ .

$x (x + 4) (x - 4) (x^{2} + 1) + 3 (- x^{4} + 15 x^{2} + 16) = 0$

Step 1.10

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x (x + 4) (x - 4) (x^{2} + 1) + 3 (- {(x^{2})}^{2} + 15 x^{2} + 16) = 0$

Step 1.11

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x (x + 4) (x - 4) (x^{2} + 1) + 3 (- u^{2} + 15 u + 16) = 0$

Step 1.12

Factor by grouping.

Tap for more steps...

Step 1.12.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 16 = - 16$ and whose sum is $b = 15$ .

Tap for more steps...

Step 1.12.1.1

Factor $15$ out of $15 u$ .

$x (x + 4) (x - 4) (x^{2} + 1) + 3 (- u^{2} + 15 (u) + 16) = 0$

Step 1.12.1.2

Rewrite $15$ as $- 1$ plus $16$

$x (x + 4) (x - 4) (x^{2} + 1) + 3 (- u^{2} + (- 1 + 16) u + 16) = 0$

Step 1.12.1.3

Apply the distributive property.

$x (x + 4) (x - 4) (x^{2} + 1) + 3 (- u^{2} - 1 u + 16 u + 16) = 0$

Step 1.12.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.12.2.1

Group the first two terms and the last two terms.

$x (x + 4) (x - 4) (x^{2} + 1) + 3 ((- u^{2} - 1 u) + 16 u + 16) = 0$

Step 1.12.2.2

Factor out the greatest common factor (GCF) from each group.

$x (x + 4) (x - 4) (x^{2} + 1) + 3 (u (- u - 1) - 16 (- u - 1)) = 0$

Step 1.12.3

Factor the polynomial by factoring out the greatest common factor, $- u - 1$ .

$x (x + 4) (x - 4) (x^{2} + 1) + 3 ((- u - 1) (u - 16)) = 0$

Step 1.13

Replace all occurrences of $u$ with $x^{2}$ .

$x (x + 4) (x - 4) (x^{2} + 1) + 3 ((- x^{2} - 1) (x^{2} - 16)) = 0$

Step 1.14

Rewrite $16$ as $4^{2}$ .

$x (x + 4) (x - 4) (x^{2} + 1) + 3 ((- x^{2} - 1) (x^{2} - 4^{2})) = 0$

Step 1.15

Factor.

Tap for more steps...

Step 1.15.1

Factor.

Tap for more steps...

Step 1.15.1.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$x (x + 4) (x - 4) (x^{2} + 1) + 3 ((- x^{2} - 1) ((x + 4) (x - 4))) = 0$

Step 1.15.1.2

Remove unnecessary parentheses.

$x (x + 4) (x - 4) (x^{2} + 1) + 3 ((- x^{2} - 1) (x + 4) (x - 4)) = 0$

Step 1.15.2

Remove unnecessary parentheses.

$x (x + 4) (x - 4) (x^{2} + 1) + 3 (- x^{2} - 1) (x + 4) (x - 4) = 0$

Step 1.16

Factor $(x + 4) (x - 4)$ out of $x (x + 4) (x - 4) (x^{2} + 1) + 3 (- x^{2} - 1) (x + 4) (x - 4)$ .

Tap for more steps...

Step 1.16.1

Factor $(x + 4) (x - 4)$ out of $x (x + 4) (x - 4) (x^{2} + 1)$ .

$(x + 4) (x - 4) ((x) (x^{2} + 1)) + 3 (- x^{2} - 1) (x + 4) (x - 4) = 0$

Step 1.16.2

Factor $(x + 4) (x - 4)$ out of $3 (- x^{2} - 1) (x + 4) (x - 4)$ .

$(x + 4) (x - 4) ((x) (x^{2} + 1)) + (x + 4) (x - 4) (3 (- x^{2} - 1)) = 0$

Step 1.16.3

Factor $(x + 4) (x - 4)$ out of $(x + 4) (x - 4) ((x) (x^{2} + 1)) + (x + 4) (x - 4) (3 (- x^{2} - 1))$ .

$(x + 4) (x - 4) ((x) (x^{2} + 1) + 3 (- x^{2} - 1)) = 0$

Step 1.17

Apply the distributive property.

$(x + 4) (x - 4) (x \cdot x^{2} + x \cdot 1 + 3 (- x^{2} - 1)) = 0$

Step 1.18

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 1.18.1

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 1.18.1.1

Raise $x$ to the power of $1$ .

$(x + 4) (x - 4) (x \cdot x^{2} + x \cdot 1 + 3 (- x^{2} - 1)) = 0$

Step 1.18.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x + 4) (x - 4) (x^{1 + 2} + x \cdot 1 + 3 (- x^{2} - 1)) = 0$

Step 1.18.2

Add $1$ and $2$ .

$(x + 4) (x - 4) (x^{3} + x \cdot 1 + 3 (- x^{2} - 1)) = 0$

Step 1.19

Multiply $x$ by $1$ .

$(x + 4) (x - 4) (x^{3} + x + 3 (- x^{2} - 1)) = 0$

Step 1.20

Apply the distributive property.

$(x + 4) (x - 4) (x^{3} + x + 3 (- x^{2}) + 3 \cdot - 1) = 0$

Step 1.21

Multiply $- 1$ by $3$ .

$(x + 4) (x - 4) (x^{3} + x - 3 x^{2} + 3 \cdot - 1) = 0$

Step 1.22

Multiply $3$ by $- 1$ .

$(x + 4) (x - 4) (x^{3} + x - 3 x^{2} - 3) = 0$

Step 1.23

Reorder terms.

$(x + 4) (x - 4) (x^{3} - 3 x^{2} + x - 3) = 0$

Step 1.24

Factor.

Tap for more steps...

Step 1.24.1

Rewrite $x^{3} - 3 x^{2} + x - 3$ in a factored form.

Tap for more steps...

Step 1.24.1.1

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.24.1.1.1

Group the first two terms and the last two terms.

$(x + 4) (x - 4) ((x^{3} - 3 x^{2}) + x - 3) = 0$

Step 1.24.1.1.2

Factor out the greatest common factor (GCF) from each group.

$(x + 4) (x - 4) (x^{2} (x - 3) + 1 (x - 3)) = 0$

Step 1.24.1.2

Factor the polynomial by factoring out the greatest common factor, $x - 3$ .

$(x + 4) (x - 4) ((x - 3) (x^{2} + 1)) = 0$

Step 1.24.2

Remove unnecessary parentheses.

$(x + 4) (x - 4) (x - 3) (x^{2} + 1) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 4 = 0$

$x - 4 = 0$

$x - 3 = 0$

$x^{2} + 1 = 0$

Step 3

Set $x + 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 3.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 4

Set $x - 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 5

Set $x - 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 6

Set $x^{2} + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 6.1

Set $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Step 6.2

Solve $x^{2} + 1 = 0$ for $x$ .

Tap for more steps...

Step 6.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 6.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 1}$

Step 6.2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 6.2.4

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 6.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Step 6.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

Step 6.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

Step 7

The final solution is all the values that make $(x + 4) (x - 4) (x - 3) (x^{2} + 1) = 0$ true.

$x = - 4, 4, 3, i, - i$

Step 8